Apostol Calculus problems

anyway I did some Apostol problems and was wondering how the Triangle Inequality and Cauchy-Schwarz are used in advanced mathematics or if they are mostly for vector algebra.
in this problem of the rectangle |x|+|y|=1 with a different norm definition are there some linear algebra theorems that touch on this. I guess I'm studying Strang's book but worry it's not abstract enough?

Re: Apostol Calculus problems

Hey mathnerd15.

They are used all the time in convergence proofs for a variety of different problems. If you can relate say a series or a sequence to a norm then you can use the results to show that something is bounded and thus converges.

This is particularly useful when you look at Hilbert spaces which are infinite-dimensional vector spaces. In finite dimensions convergence is obvious but in infinite dimensions, you get a lot of problems since many properties of convergence in finite spaces don't necessarily work in hilbert spaces.

Also note that you can apply norms to integrals since they are just an infinite summation as well.